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Register a new userProblems from the workshop on Automorphisms of Curves (Leiden, August, 2004)
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Contents: Rational functions with given monodromy on generic curves (I. Bouw & S. Wewers); Can deformation rings of group representations not be local complete intersections? (T. Chinburg); Lifting an automorphism group to finite characteristic (G. Cornelissen); Flat connections and representations of the fundamental group in characteristic p>0 (C. Gasbarri) Questions on p-torsion of hyperelliptic curves (D. Glass & R. Pries); Automorphisms of curves and stable reduction (C. Lehr & M. Matignon); p-groups as automorphism groups of curves in characteristic p; Lifting Galois covers of smooth curves (M. Matignon); Abelian varieties isogenous to a Jacobian (F. Oort); Minimal maximal number of automorphisms of curves (F. Oort)
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Let ${cal M}_{g,[n]}$, for $2g-2+n>0$, be the D-M moduli stack of smooth curves of genus $g$ labeled by $n$ unordered distinct points. The main result of the paper is that a finite, connected etale cover ${cal M}^l$ of ${cal M}_{g,[n]}$, defined over a sub-$p$-adic field $k$, is almost anabelian in the sense conjectured by Grothendieck for curves and their moduli spaces. The precise result is the following. Let $pi_1({cal M}^l_{ol{k}})$ be the geometric algebraic fundamental group of ${cal M}^l$ and let ${Out}^*(pi_1({cal M}^l_{ol{k}}))$ be the group of its exterior automorphisms which preserve the conjugacy classes of elements corresponding to simple loops around the Deligne-Mumford boundary of ${cal M}^l$ (this is the $ast$-condition motivating the almost above). Let us denote by ${Out}^*_{G_k}(pi_1({cal M}^l_{ol{k}}))$ the subgroup consisting of elements which commute with the natural action of the absolute Galois group $G_k$ of $k$. Let us assume, moreover, that the generic point of the D-M stack ${cal M}^l$ has a trivial automorphisms group. Then, there is a natural isomorphism: $${Aut}_k({cal M}^l)cong{Out}^*_{G_k}(pi_1({cal M}^l_{ol{k}})).$$ This partially extends to moduli spaces of curves the anabelian properties proved by Mochizuki for hyperbolic curves over sub-$p$-adic fields.
This paper describes a class of Artin-Schreier curves, generalizing results of Van der Geer and Van der Vlugt to odd characteristic. The automorphism group of these curves contains a large extraspecial group as a subgroup. Precise knowledge of this subgroup makes it possible to compute the zeta functions of the curves in the class over the field of definition of all automorphisms in the subgroup. As a consequence, we obtain new examples of maximal curves.
We investigate degenerations of syzygy bundles on plane curves over $p$-adic fields. We use Mustafin varieties which are degenerations of projective spaces to find a large family of models of plane curves over the ring of integers such that the special fiber consists of multiple projective lines meeting in one point. On such models we investigate vector bundles whose generic fiber is a syzygy bundle and which become trivial when restricted to each projective line in the special fiber. Hence these syzygy bundles have strongly semistable reduction. This investigation is motivated by the fundamental open problem in $p$-adic Simpson theory to determine the category of Higgs bundles corresponding to continuous representations of the etale fundamental group of a curve. Faltings $p$-adic Simpson correspondence and work of Deninger and the second author shows that bundles with Higgs field zero and potentially strongly semistable reduction fall into this category. Hence the results in the present paper determine a class of syzygy bundles on plane curves giving rise to a $p$-adic local system. We apply our methods to a concrete example on the Fermat curve suggested by Brenner and prove that this bundle has potentially strongly semistable reduction.
We prove equidistribution of Weierstrass points on Berkovich curves. Let $X$ be a smooth proper curve of positive genus over a complete algebraically closed non-Archimedean field $K$ of equal characteristic zero with a non-trivial valuation. Let $L$ be a line bundle of positive degree on $X$. The Weierstrass points of powers of $L$ are equidistributed according to the Zhang-Arakelov measure on the analytification $X^{an}$. This provides a non-Archimedean analogue of a theorem of Mumford and Neeman. Along the way we provide a description of the reduction of Weierstrass points, answering a question of Eisenbud and Harris.
Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C:y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over K, and $J$ the jacobian of $C$. We identify $C$ with the image of its canonical embedding into $J$ (the infinite point of $C$ goes to the identity element of $J$). It is well known that for each $mathfrak{b} in J(K)$ there are exactly $2^{2g}$ elements $mathfrak{a} in J(K)$ such that $2mathfrak{a}=mathfrak{b}$. M. Stoll constructed an algorithm that provides Mumford representations of all such $mathfrak{a}$, in terms of the Mumford representation of $mathfrak{b}$. The aim of this paper is to give explicit formulas for Mumford representations of all such $mathfrak{a}$, when $mathfrak{b}in J(K)$ is given by $P=(a,b) in C(K)subset J(K)$ in terms of coordinates $a,b$. We also prove that if $g>1$ then $C(K)$ does not contain torsion points with order between $3$ and $2g$.
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